Key words and phrases: Hilbert’s integral inequality

http://jipam.vu.edu.au/

Volume 6, Issue 2, Article 39, 2005

ON A NEW MULTIPLE EXTENSION OF HILBERT’S INTEGRAL INEQUALITY

BICHENG YANG DEPARTMENT OFMATHEMATICS, GUANGDONGINSTITUTE OFEDUCATION,

GUANGZHOU, GUANGDONG510303, PEOPLE’SREPUBLICOFCHINA. bcyang@pub.guangzhou.gd.cn

URL:http://page.gdei.edu.cn/ yangbicheng

Received 04 June, 2004; accepted 18 March, 2005 Communicated by B.G. Pachpatte

ABSTRACT. This paper gives a new multiple extension of Hilbert’s integral inequality with a best constant factor, by introducing a parameterλand theΓfunction. Some particular results are obtained.

Key words and phrases: Hilbert’s integral inequality; Weight coefficient ,Γfunction.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Iff, g ≥0satisfy 0<

Z ∞ 0

f²(x)dx <∞ and 0<

Z ∞ 0

g²(x)dx <∞,

then (1.1)

Z ∞ 0

Z ∞ 0

f(x)g(y)

x+y dxdy < π Z ∞

0

f²(x)dx Z ∞

0

g²(x)dx ¹₂

,

where the constant factor π is the best possible (cf. Hardy et al. [2]). Inequality (1.1) is well known as Hilbert’s integral inequality, which had been extended by Hardy [1] as:

Ifp > 1, ¹_p +¹_q = 1, f, g ≥0satisfy 0<

Z ∞ 0

f^p(x)dx <∞ and 0<

Z ∞ 0

g^q(x)dx <∞,

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

144-04

~

then (1.2)

Z ∞ 0

Z ∞ 0

f(x)g(y)

x+y dxdy < π sin

π p

Z ∞

0

f^p(x)dx

¹_pZ ∞ 0

g^q(x)dx ¹_q

,

where the constant factor _sin(π/p)^π is the best possible. Inequality (1.2) is called Hardy- Hilbert’s integral inequality, and is important in analysis and its applications (cf. Mitrinovi´c et al.[6]).

Recently, by introducing a parameterλ,Yang [9] gave an extension of (1.2) as:

Ifλ >2−min{p, q}, f, g ≥0satisfy 0<

Z ∞ 0

x^1−λf^p(x)dx <∞ and 0<

Z ∞ 0

x^1−λg^q(x)dx <∞,

then (1.3)

Z ∞ 0

Z ∞ 0

f(x)g(y)

(x+y)^λdxdy < k_λ(p) Z ∞

0

x^1−λf^p(x)dx

¹_pZ ∞ 0

x^1−λg^q(x)dx ¹_q

, where the constant factor k_λ(p) = B_p+λ−2

p ,^q+λ−2_q

is the best possible (B(u, v) is the β function). Forλ= 1,inequality (1.3) reduces to (1.2).

On the problem for multiple extension of (1.1), [3, 4] gave some new results and Yang [8]

gave an improvement of their works as:

Ifn ∈N\{1}, p_i >1,Pn i=1

1

pi = 1, λ > n− min

1≤i≤n{p_i}, f_i ≥0,satisfy 0<

Z ∞ 0

x^n−1−λf_i^pi(x)dx <∞ (i= 1,2, . . . , n),

then (1.4)

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fi(xi)dx1. . . dxn

< 1 Γ(λ)

n

Y

i=1

Γ

p_i+λ−n p_i

Z ∞ 0

x^n−1−λf_i^pi(x)dx _pi¹

,

where the constant factor_Γ(λ)¹ Qn i=1Γ

pi+λ−n pi

is the best possible. Forn= 2, inequality (1.4) reduces to (1.3). It follows that (1.4) is a multiple extension of (1.3), (1.2) and (1.1).

In 2003, Yang et. al [11] provided an extensive account of the above results.

The main objective of this paper is to build a new extension of (1.1) with a best constant factor other than (1.4), and give some new particular results. That is

Theorem 1.1. Ifn∈N\{1}, p_i >1,Pn i=1

1

pi = 1, λ >0, f_i ≥0,satisfy 0<

Z ∞ 0

x^pi−1−λf_i^pi(x)dx <∞ (i= 1,2, . . . , n),

then (1.5)

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

f_i(x_i)dx₁. . . dx_n

< 1 Γ(λ)

n

Y

i=1

Γ λ

pi

Z ∞ 0

x^pi−1−λf_i^pi(x)dx _pi¹

,

where the constant factor _Γ(λ)¹ Qn i=1Γ

λ pi

is the best possible. In particular,

(a) forλ= 1,we have (1.6)

Z ∞ 0

· · · Z ∞

0

Qn

i=1f_i(x_i) Pn

j=1xj

dx₁. . . dx_n <

n

Y

i=1

Γ 1

pi

Z ∞ 0

x^pi−2f_i^pi(x)dx _pi¹

;

(b) forn= 2, using the symbol of (1.3) and settingek_λ(p) = B

λ p,^λ_q

,we have

(1.7) Z ∞

0

Z ∞ 0

f(x)g(y)

(x+y)^λdxdy <ekλ(p) Z ∞

0

x^p−1−λf^p(x)dx

¹_pZ ∞ 0

x^q−1−λg^q(x)dx ¹_q

,

where the constant factors in (1.6) and (1.7) are still the best possible.

In order to prove the theorem, we introduce some lemmas.

2. SOME LEMMAS

Lemma 2.1. Ifk ∈N, r_i >1 (i= 1,2, . . . , k+ 1),andPk+1

i=1 r_i =λ(k),then

(2.1)

Z ∞ 0

· · · Z ∞

0

1

1 +Pk

j=1u_jλ(k) k

Y

j=1

u^r_j^j−1du₁. . . du_k = Qk+1

i=1 Γ(r_i) Γ(λ(k)) .

Proof. We establish (2.1) by mathematical induction. Fork = 1, sincer₁+r₂ =λ(1),and (see [7])

(2.2) B(p, q) =

Z ∞ 0

u^p−1

(1 +u)^p+qdu=B(q, p) (p, q >0),

we have (2.1). Suppose for k ∈ N,that (2.1) is valid. Then fork + 1, sincer₁ +Pk+1 i=2 r_i = λ(k+ 1), by settingv =u₁.

1 +Pk+1 j=2u_j

, we obtain Z ∞

0

· · · Z ∞

0

1

1 +Pk+1

j=1u_jλ(k+1) k+1

Y

j=1

u^r_j^j−1du₁. . . du_k+1 (2.3)

= Z ∞

0

· · · Z ∞

0

v^r1−1

1 +Pk+1 j=2u_jr1

Qk+1 j=2u^r_j^j−1

1 +Pk+1

j=2u_jλ(k+1)

(1 +v)^λ(k+1)

dvdu₂. . . du_k+1

= Z ∞

0

· · · Z ∞

0

Qk+1 j=2u^r_j^j−1

1 +Pk+1

j=2 u_jλ(k+1)−r1du₂. . . du_k+1 Z ∞

0

v^r1−1

(1 +v)^λ(k+1)dv.

In view of (2.2) and the assumption ofk, we have (2.4)

Z ∞ 0

v^r1−1

(1 +v)^λ(k+1)dv = 1

Γ(λ(k+ 1))Γ

k+1

X

i=2

r_i

! Γ(r₁);

(2.5)

Z ∞ 0

· · · Z ∞

0

Qk+1 j=2u^r_j^j−1

1 +Pk+1

j=2u_jλ(k+1)−r1du₂. . . du_k+1 =

Qk+2 i=2 Γ(r_i) Γ

Pk+1 i=2 r_i.

Then, by (2.5), (2.4) and (2.3), we find Z ∞

0

· · · Z ∞

0

Qk+1 j=1u^r_j^j−1

1 +Pk+1

j=1u_jλ(k+1)du₁. . . du_k+1 = Qk+2

i=1 Γ(r_i) Γ(λ(k+ 1)).

Hence (2.1) is valid fork ∈Nby induction. The lemma is proved.

Lemma 2.2. Ifn ∈ N\{1}, p_i > 1 (i = 1,2, . . . , n),Pn i=1

1

pi = 1andλ > 0, set the weight coefficientω(x_i)as

(2.6) ω(x_i) :=x

λ pj

i

Z ∞ 0

· · · Z ∞

0

Qn

j=1(j6=i)x^(λ−p_j ^j)/pj Pn

j=1x_jλ dx₁. . . dx_i−1dx_i+1. . . dx_n. Then, eachω(x_i)is constant, that is

(2.7) ω(x_i) = 1

Γ(λ)

n

Y

j=1

Γ λ

pj

, (i= 1,2, . . . , n).

Proof. Fixi. Settingpe_n =p_i,andpe_j =p_j, u_j =x_j/x_i, forj = 1,2, . . . , i−1;pe_j =p_j+1, u_j = x_j+1/x_i, forj =i, i+ 1, . . . , n−1in (2.6), by simplification, we have

(2.8) ω(x_i) = Z ∞

0

· · · Z ∞

0

1

1 +Pn−1 j=1 u_jλ

n−1

Y

j=1

u

−1+^λ

pj

j du₁. . . dun−1.

Substitution ofn−1fork, λforλ(k)andλ/pej forrj (j = 1,2, . . . , n)into (2.1), in view of (2.8), we have

ω(x_i) = 1 Γ(λ)

n

Y

j=1

Γ λ

pe_j

= 1

Γ(λ)

n

Y

j=1

Γ λ

p_j

.

Hence, (2.7) is valid. The lemma is proved.

Lemma 2.3. As in the assumption of Lemma 2.2, for0< ε < λ, we have

I :=ε Z ∞

1

· · · Z ∞

1

Qn

i=1x^(λ−p_i ^i−ε)/pi Pn

j=1x_jλ dx1. . . dxn

≥ 1

Γ(λ)

n

Y

i=1

Γ λ

p_i

(ε→0⁺).

(2.9)

Proof. Settingu_i =x_i/x_n(i= 1,2, . . . , n−1)in the following, we find I =ε

Z ∞ 1

x^−1−ε_n





 Z ∞

x⁻¹n

· · · Z ∞

x⁻¹n

Qn−1

i=1 u^(λ−p_i ^i−ε)/pi

1 +Pn−1

j=1u_jλdu1. . . dun−1





dxn

≥ε Z ∞

1

x^−1−ε_n





 Z ∞

0

· · · Z ∞

0

Qn−1

i=1 u^(λ−p_i ^i−ε)/pi

1 +Pn−1 j=1 uj

λdu₁. . . du_n−1





dx_n

−ε Z ∞

1

x⁻¹_n

n−1

X

j=1

A_j(x_n)dx_n, (2.10)

where, forj = 1,2, . . . , n−1, A_j(x_n)is defined by

(2.11) A_j(x_n) :=

Z

· · · Z

Dj

Qn−1

i=1 u^(λ−p_i ^i−ε)/pi (1 +Pn−1

j=1 u_j)^λ du₁. . . dun−1, satisfyingD_j ={(u₁, u₂, . . . , un−1)|0< u_j ≤x⁻¹_n , 0< u_k<∞(k6=j)}.

Without loss of generality, we estimate the integralAj(xn)forj = 1.

(a) Forn = 2, we have A₁(x_n) =

Z x⁻¹n

0

1

(1 +u₁)^λu^(λ−p₁ ^1−ε)/p1du₁

≤ Z x⁻¹n

0

u^(λ−p₁ ^1−ε)/p1du₁ = p1

λ−εx^{−(λ−ε)/p}_n ¹;

(b) Forn ∈N\{1,2},by (2.1), we have

A₁(x_n)≤ Z ∞

0

· · · Z ∞

0

Qn−1 i=2 u⁻¹⁺

λ−ε pi

i

1 +Pn−1

j=2u_jλdu₁. . . dun−1

Z x⁻¹n

0

u

λ−p1−ε p1

1 du₁

≤ p₁x⁻

λ−ε p1

n

λ−ε Z ∞

0

· · · Z ∞

0

Qn−1

i=2 u^{−1+(λ−ε)/p}_i ⁱ

1 +Pn−1 j=2 uj

(λ−ε)(1−p⁻¹₁ )du₁. . . du_n−1

= p₁x⁻

λ−ε p1

n

λ−ε ·

Qn

i=2Γ(^λ−ε_p

i ) Γ((λ−ε)(1−p⁻¹₁ )).

By virtue of the results of (a) and (b), forj = 1,2, . . . , n−1,we have (2.12) A_j(x_n)≤ p_j

λ−εx^{−(λ−ε)/p}_n ^jO_j(1) (ε →0⁺, n∈N\{1}).

By (2.11), since forε →0⁺, Z ∞

0

· · · Z ∞

0

Qn−1

i=1 u^{−1+(λ−ε)/p}_i ⁱ

1 +Pn−1

j=1u_jλ du₁. . . dun−1 = Qn

i=1Γ

λ pi

Γ(λ) +o(1);

Z ∞ 1

x⁻¹_n

n−1

X

j=1

A_j(x_n)dx_n=

n−1

X

j=1

Z ∞ 1

x⁻¹_n A_j(x_n)dx_n

≤

n−1

X

j=1

p_j

λ−εO_j(1) Z ∞

1

x^{−1−(λ−ε)/p}_n ^jdx_n

=

n−1

X

j=1

p_j λ−ε

2

O_j(1),

then by (2.10) , we find I ≥



 Qn

i=1Γ

λ pi

Γ(λ) +o(1)



−ε

n−1

X

j=1

p_j λ−ε

2

O_j(1)

→ Qn

i=1Γ λ

pi

Γ(λ) (ε→0⁺).

Thereby, (2.9) is valid and the lemma is proved.

3. PROOF OF THE THEOREM ANDREMARKS

Proof of Theorem 1.1. By Hölder’s inequality, we have J :=

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1xj

λ n

Y

i=1

f_i(x_i)dx₁. . . dx_n

= Z ∞

0

· · · Z ∞

0











n

Y

i=1

f_i(x_i) Pn

j=1x_jλ/pi









x^{(pi−λ)(1−p}

−1 i ) i

n

Y

j=1

(j6=i)

x

λ−pj pj

j









1 pi











dx₁. . . dx_n

≤

n

Y

i=1









 Z ∞

0

· · · Z ∞

0

f_i^pi(x_i) Pn

j=1x_jλx^{(pi−λ)(1−p}

−1 i ) i

n

Y

j=1

(j6=i)

x

λ−pj pj

j dx1. . . dxn











1 pi

. (3.1)

If (3.1) takes the form of equality, then there exists constantsC₁, C₂, . . . , C_n,such that they are not all zero and for anyi6=k∈ {1,2, . . . , n}(see [5]),

Ci

f_i^pi(x_i)x^{(pi−λ)(1−p}

−1 i ) i

Pn

j=1x_jλ

n

Y

j=1

(j6=i)

x

λ−pj pj

j =Ck

f_k^pk(x_k)x^{(pk−λ)(1−p}

−1 k ) k

Pn

j=1x_jλ

n

Y

j=1

(j6=k)

x

λ−pj pj

j ,

a.e. in(0,∞)× · · · ×(0,∞).

(3.2)

Assume thatC_i 6= 0.By simplification of (3.2), we find

x^p_i^i−λf_i^pi(x_i) = F(x₁, . . . , xi−1, x_i+1, . . . , x_n)

=constant a.e. in(0,∞)× · · · ×(0,∞), which contradicts the fact that0 < R∞

0 x^p_i^i−λ−1f_i^pi(x)dx < ∞.Hence by (2.6) and (3.1), we conclude

(3.3) J <

n

Y

i=1

Z ∞ 0

ω(x_i)x^p_i^i−1−λf_i^pi(x_i)dx_{i pi}¹

. Then by (2.7), we have (1.5).

For0< ε < λ,settingfe_i(x_i)as: fe_i(x_i) = 0,forx_i ∈(0,1);

fe_i(x_i) = x^(λ−p_i ^i−ε)/pi, forx_i ∈[1,∞) (i= 1,2, . . . , n),

then we find

(3.4) ε

n

Y

i=1

Z ∞ 0

x^p_i^i−1−λfe_i^pi(x_i)dx_i ¹

pi = 1.

By (2.9), we find

(3.5) ε

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fei(xi)dx1. . . dxn

=I ≥ 1 Γ(λ)

n

Y

i=1

Γ λ

p_i

(ε→0⁺).

If the constant factor_Γ(λ)¹ Qn i=1Γ

λ pi

in (1.5) is not the best possible, then there exists a posi- tive constantK < _Γ(λ)¹ Qn

i=1Γ

λ pi

, such that (1.5) is still valid if one replaces_Γ(λ)¹ Qn i=1Γ

λ pi

byK.In particular, one has

ε Z ∞

0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fe_i(x_i)dx₁. . . dx_n< εK

n

Y

i=1

Z ∞ 0

x^pi−1−λfe_i^pi(x)dx ¹

pi ,

and in view of (3.4) and (3.5), it follows that _Γ(λ)¹ Qn i=1Γ

λ pi

≤K(ε→0⁺).This contradicts the factK < _Γ(λ)¹ Qn

i=1Γ

λ pi

.Hence the constant factor _Γ(λ)¹ Qn i=1Γ

λ pi

in (1.5) is the best possible.

The theorem is proved.

Remark 3.1. Forλ= 1, inequality (1.7) reduces to (see [10])

(3.6)

Z ∞ 0

Z ∞ 0

f(x)g(y)

x+y dxdy < π sin

π p

Z ∞

0

x^p−2f^p(x)dx

_p¹ Z ∞ 0

x^q−2g^q(x)dx ¹_q

. Forp =q = 2, both (3.6) and (1.2) reduce to (1.1). It follows that inequalities (3.6) and (1.2) are different extensions of (1.1). Hence, inequality (1.5) is a new multiple extension of (1.1).

Since all the constant factors in the obtained inequalities are the best possible, we have obtained new results.

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